Abstract: In the paper [X. Tan et al., Phys. Rev. Lett. 122, 210401 (2019)], we have
studied the Euler characteristic number $\chi$ for a two-band model; However,
the calculated $\chi$ there is not an integer when the parameter $|h|>1$ and
then may not be considered as a suitable topological number for the system. In
this note, we find that the Bloch vectors of the ground state for $|h|>1$ do
not cover the whole Bloch sphere and thus the Euler characteristic number
should be effectively associated with the manifold of a disk, rather than a
sphere. After taking into account the boundary contribution, we derive the
correct Euler characteristic number. Unfortunately, the Euler characteristic
number does not change when crossing the critical points $|h|=1$ and thus can
not be used to characterize the topological phase transition of the present
model.